Text: EconS 527
Problem Set 2. Technology and profit maximization
Problem 1. For the following production functions, derive the elasticity of output of capital (K) and labor (L), technical rate of substitution between K and L, elasticity of substitution between K and L, and elasticity of scale.
Q = baK + (1-a)L/r
bO = aK + bL + cE
CQ = KLFB
Problem 2. For the following profit maximization problem, derive the input demand functions, supply function, and profit function. Show the first and second order conditions. Determine how p affects input demand x, output supply, and profit.
(amaxplnx - wx) where w = ax
For the following profit maximization problem, derive the determinants of optimal output by solving the first order conditions correctly. Show the conditions verifying that the agent maximizes profit. Write the profit function. Determine how p affects outputs z, y, and profit. Assume Cy > 0.
(bmaxpz + qy - Cz, y)
Problem 3. The following production function characterizes production of good y.
f(x, z) = x^a * z^b where 0 < a < 1 and 0 < b < 1.
The output price is p. The input prices for x and z are w and wz respectively.
Set up the problem for a profit maximizing firm and solve for both input demand functions. Do not forget to show the first order condition and show if the second order condition satisfies the condition for a maximum. Derive the profit function. What is the derivative of the profit function with respect to w? Show the expression and interpret it.
Problem 4. A firm has the following general production function f(M, E) where M is machinery and E is energy. The firm produces output y using this production function at price p, and the price of machinery and energy are r and e, respectively. The government taxes the firm t for every unit of output sold and subsidizes the firm s for each unit of energy purchased.
a. Set up the problem for a profit maximizing firm. Derive the first and second order conditions. Write the input demand functions as functions of all relevant parameters.
b. How does a decrease in the subsidy affect demand for machinery and profit? Make any necessary assumptions you need. Prove your answer. How does an increase in the tax affect demand for energy and profit? Make any necessary assumptions you need. Prove your answer.
Deadline: September 20, 2019, 2pm. See syllabus for penalty due to late submissions.